This chapter will teach you how one can cluster and classify datasets. There is different methods to perform the clustering of the data, but in a very simple point of view you cluster what is similar. For that you can use different parameters, from simple concepts of means to more complex concepts like data populations distances.
The data set Boston is included in the library {MASS} and refers to the “Housing Values in Suburbs of Boston”. The data compiled in the dataset orginates from the following papers: Harrison and Rubinfeld (1978) & Belsley et al 1980
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The Boston data set includes the Housing Values in Suburbs of Boston (from 1978 - 1980). The information included in this data set includes values regarding Boston city crime, nitrogen oxide concentrations, average rooms per dwelling, lower status of the population, pupil-teacher ratio in town, etc..
## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...
## [1] 506 14
You can see that the data includes a total of 506 observations distributed into the following 14 variables: - crim = per capita crime rate by town. - zn = proportion of residential land zoned for lots over 25,000 sq.ft. - indus = proportion of non-retail business acres per town. - chas = Charles River dummy variable (= 1 if tract bounds river; 0 otherwise). - nox = nitrogen oxides concentration (parts per 10 million). - rm = average number of rooms per dwelling. - age = proportion of owner-occupied units built prior to 1940. - dis = weighted mean of distances to five Boston employment centres. - rad = index of accessibility to radial highways. - tax = full-value property-tax rate per $10,000. - ptratio = pupil-teacher ratio by town. - black = 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town. - lstat = lower status of the population (percent). - medv = median value of owner-occupied homes in $1000s.
## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00
## corrplot 0.84 loaded
## crim zn indus chas nox rm age dis rad tax
## crim 1.00 -0.20 0.41 -0.06 0.42 -0.22 0.35 -0.38 0.63 0.58
## zn -0.20 1.00 -0.53 -0.04 -0.52 0.31 -0.57 0.66 -0.31 -0.31
## indus 0.41 -0.53 1.00 0.06 0.76 -0.39 0.64 -0.71 0.60 0.72
## chas -0.06 -0.04 0.06 1.00 0.09 0.09 0.09 -0.10 -0.01 -0.04
## nox 0.42 -0.52 0.76 0.09 1.00 -0.30 0.73 -0.77 0.61 0.67
## rm -0.22 0.31 -0.39 0.09 -0.30 1.00 -0.24 0.21 -0.21 -0.29
## age 0.35 -0.57 0.64 0.09 0.73 -0.24 1.00 -0.75 0.46 0.51
## dis -0.38 0.66 -0.71 -0.10 -0.77 0.21 -0.75 1.00 -0.49 -0.53
## rad 0.63 -0.31 0.60 -0.01 0.61 -0.21 0.46 -0.49 1.00 0.91
## tax 0.58 -0.31 0.72 -0.04 0.67 -0.29 0.51 -0.53 0.91 1.00
## ptratio 0.29 -0.39 0.38 -0.12 0.19 -0.36 0.26 -0.23 0.46 0.46
## black -0.39 0.18 -0.36 0.05 -0.38 0.13 -0.27 0.29 -0.44 -0.44
## lstat 0.46 -0.41 0.60 -0.05 0.59 -0.61 0.60 -0.50 0.49 0.54
## medv -0.39 0.36 -0.48 0.18 -0.43 0.70 -0.38 0.25 -0.38 -0.47
## ptratio black lstat medv
## crim 0.29 -0.39 0.46 -0.39
## zn -0.39 0.18 -0.41 0.36
## indus 0.38 -0.36 0.60 -0.48
## chas -0.12 0.05 -0.05 0.18
## nox 0.19 -0.38 0.59 -0.43
## rm -0.36 0.13 -0.61 0.70
## age 0.26 -0.27 0.60 -0.38
## dis -0.23 0.29 -0.50 0.25
## rad 0.46 -0.44 0.49 -0.38
## tax 0.46 -0.44 0.54 -0.47
## ptratio 1.00 -0.18 0.37 -0.51
## black -0.18 1.00 -0.37 0.33
## lstat 0.37 -0.37 1.00 -0.74
## medv -0.51 0.33 -0.74 1.00
By drawing the correlation plot you can see some variables are highly positive or negatively correlated with each other.
High positive numbers (Blue) mean that variables value will shift together in the same direction. For example, when you look at the crime data, you can see its higly positvely correlated with the tax, rad and lstat variables. Meaning that the higher the index of acessibility to radial highways, the higher value of property-tax rate per $10.000, and the amount of low status of the population the bigger amount of crime you can witness on those areas.
On the other side, you can also observe some High negative numbers (Red) meaning that while the value of one variable increases the other one will decrease. For example, you can see how lstat (percentage of the lower status of the population) inversely correlates with medv (median value of owner-occupied homes in $1000s) and rm (average number of rooms per dwelling). Meaning that the higher amount of low status population the less amount of rooms per dwelling, and the less amount of owner occupied homes with the value of $1000.
## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865
## [1] "matrix"
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.419367 -0.410563 -0.390280 0.000000 0.007389 9.924110
## 0% 25% 50% 75% 100%
## -0.419366929 -0.410563278 -0.390280295 0.007389247 9.924109610
## crime
## low med_low med_high high
## 127 126 126 127
## # A tibble: 506 x 14
## # Groups: crime [4]
## zn indus chas nox rm age
## * <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 0.28454827 -1.2866362 -0.2723291 -0.1440749 0.4132629 -0.11989477
## 2 -0.48724019 -0.5927944 -0.2723291 -0.7395304 0.1940824 0.36680343
## 3 -0.48724019 -0.5927944 -0.2723291 -0.7395304 1.2814456 -0.26554897
## 4 -0.48724019 -1.3055857 -0.2723291 -0.8344581 1.0152978 -0.80908783
## 5 -0.48724019 -1.3055857 -0.2723291 -0.8344581 1.2273620 -0.51067434
## 6 -0.48724019 -1.3055857 -0.2723291 -0.8344581 0.2068916 -0.35080997
## 7 0.04872402 -0.4761823 -0.2723291 -0.2648919 -0.3880270 -0.07015919
## 8 0.04872402 -0.4761823 -0.2723291 -0.2648919 -0.1603069 0.97784057
## 9 0.04872402 -0.4761823 -0.2723291 -0.2648919 -0.9302853 1.11638970
## 10 0.04872402 -0.4761823 -0.2723291 -0.2648919 -0.3994130 0.61548134
## # ... with 496 more rows, and 8 more variables: dis <dbl>, rad <dbl>,
## # tax <dbl>, ptratio <dbl>, black <dbl>, lstat <dbl>, medv <dbl>,
## # crime <fctr>
## [1] 506
## zn indus chas nox rm
## 353 2.08538800 -1.37701059 -0.2723291 -1.24005818 -0.570202978
## 176 -0.48724019 -1.03300497 -0.2723291 -0.38570897 0.371988664
## 283 0.37030254 -1.13795583 3.6647712 -0.96476788 1.936140650
## 100 -0.48724019 -1.20209248 -0.2723291 -0.94664532 1.610216351
## 359 -0.48724019 1.01499462 3.6647712 1.85803641 -0.224353176
## 337 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.591551731
## 53 0.41317968 -0.80123846 -0.2723291 -0.99842406 0.322174907
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## 60 0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.509003218
## 15 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.268473932
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## 94 0.71331963 0.56895343 -0.2723291 -0.78267931 -0.104800158
## 56 3.37170210 -1.44552019 -0.2723291 -1.30909650 1.372533565
## 50 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.971559538
## 350 1.22784527 -1.44114723 -0.2723291 -1.08472196 0.931325998
## 449 -0.48724019 1.01499462 -0.2723291 1.36613839 -0.141804663
## 307 0.92770532 -1.30558569 -0.2723291 -0.71364099 1.615909352
## 143 -0.48724019 1.23072696 3.6647712 2.72964520 -1.254786331
## 130 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.921745781
## 21 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -1.017103545
## 140 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.190195170
## 263 0.37030254 -1.04466617 -0.2723291 0.79657225 3.007848061
## 107 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.638518988
## 244 0.79907391 -0.90473168 -0.2723291 -1.09335175 0.154231381
## 425 -0.48724019 1.01499462 -0.2723291 0.25289548 -1.024219796
## 252 0.45605682 -0.76917014 -0.2723291 -1.06746238 0.218277641
## 85 -0.48724019 -0.96886832 -0.2723291 -0.91212616 0.148538381
## 248 0.45605682 -0.76917014 -0.2723291 -1.06746238 -0.083451404
## 246 0.45605682 -0.76917014 -0.2723291 -1.06746238 -0.967289788
## 59 0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.198734672
## 322 -0.48724019 -0.54760720 -0.2723291 -0.53241540 0.130036128
## 227 -0.48724019 -0.71961001 -0.2723291 -0.43748771 2.498324485
## 340 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.426454706
## 326 -0.48724019 -0.54760720 -0.2723291 -0.53241540 0.208314890
## 377 -0.48724019 1.01499462 -0.2723291 1.00368721 0.518583436
## 346 -0.48724019 -0.98344483 -0.2723291 -0.97253469 -0.385180450
## 301 2.51415937 -1.29683979 -0.2723291 -1.33498587 0.834544984
## 291 2.94293073 -0.90181638 -0.2723291 -1.24005818 0.820312482
## 440 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.935978283
## 208 -0.48724019 -0.07970124 -0.2723291 -0.56693456 -0.713951249
## 89 -0.48724019 -1.12629463 -0.2723291 -0.56693456 1.028107013
## 380 -0.48724019 1.01499462 -0.2723291 1.00368721 -0.087721155
## 268 0.37030254 -1.04466617 -0.2723291 0.17522737 2.864099790
## 127 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.955903786
## 194 2.08538800 -1.19626187 -0.2723291 -1.32635608 0.733494219
## 148 -0.48724019 1.23072696 -0.2723291 2.72964520 -1.933676683
## 104 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.210120673
## 349 2.94293073 -1.33036576 -0.2723291 -1.03294322 0.498657933
## 429 -0.48724019 1.01499462 -0.2723291 1.07272553 -0.130418661
## 57 3.15731641 -1.51548743 -0.2723291 -1.24868797 0.139998879
## 409 -0.48724019 1.01499462 -0.2723291 0.36508275 -0.950210785
## 321 -0.48724019 -0.54760720 -0.2723291 -0.53241540 0.201198639
## 467 -0.48724019 1.01499462 -0.2723291 0.86561057 -0.473421963
## 398 -0.48724019 1.01499462 -0.2723291 1.19354259 -0.765188257
## 468 -0.48724019 1.01499462 -0.2723291 0.25289548 -0.400836202
## 134 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.658444491
## 270 0.37030254 -0.60882854 3.6647712 -0.78267931 -0.518965970
## 406 -0.48724019 1.01499462 -0.2723291 1.19354259 -0.856276271
## 305 0.92770532 -1.30558569 -0.2723291 -0.71364099 1.354031312
## 111 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.127572161
## 389 -0.48724019 1.01499462 -0.2723291 1.25395112 -1.999146193
## age dis rad tax ptratio
## 353 -1.778931675 3.2840499862 -0.6373311 0.01639310 -0.07184181
## 176 -1.260260608 -0.3147359550 -0.5224844 -0.66594918 -0.85708096
## 283 -0.670538709 0.6728643674 -0.5224844 -1.14062207 -1.64232012
## 100 -0.215813389 -0.1423950448 -0.8670245 -0.78461740 -0.21041342
## 359 0.526667797 -0.5092546567 1.6596029 1.52941294 0.80577843
## 337 -0.791325122 0.6819824315 -0.5224844 -1.09315478 0.80577843
## 53 -1.686565595 1.4340327636 -0.6373311 -0.98041997 -0.76469989
## 431 0.622586419 -0.8274371034 1.6596029 1.52941294 0.80577843
## 430 0.960077867 -0.8675660836 1.6596029 1.52941294 0.80577843
## 180 -0.361467593 -0.4587728745 -0.7521778 -1.27709053 -0.30279450
## 323 -0.663433626 0.4397838527 -0.5224844 -0.71934988 0.52863520
## 60 -0.759352248 1.4897384367 -0.1779443 -0.73715011 0.57482574
## 15 0.565745754 0.3166899868 -0.6373311 -0.60068166 1.17530274
## 95 0.309962761 -0.0855021237 -0.6373311 -0.82021787 -0.11803234
## 314 0.505352547 -0.2527616128 -0.6373311 -0.61848189 -0.02565127
## 74 -2.215894287 0.7086717651 -0.6373311 -0.61254848 0.34387304
## 285 -1.697223220 1.6679680968 -0.9818712 -0.73121670 -1.45755797
## 164 0.899684660 -0.7755306237 -0.5224844 -0.03107419 -1.73470120
## 382 1.084416821 -1.0807958128 1.6596029 1.52941294 0.80577843
## 72 -1.814457091 0.7086717651 -0.6373311 -0.61254848 0.34387304
## 494 -0.517779422 -0.6711952751 -0.4076377 -0.10227512 0.34387304
## 296 -1.331311439 1.0283263992 -0.6373311 -0.70748306 -1.13422420
## 259 1.116389695 -0.9025186628 -0.5224844 -0.85581834 -2.51994036
## 2 0.366803426 0.5566090496 -0.8670245 -0.98635338 -0.30279450
## 146 1.116389695 -1.1283332201 -0.5224844 -0.03107419 -1.73470120
## 367 0.810871121 -0.9700968153 1.6596029 1.52941294 0.80577843
## 388 0.743372832 -1.0811757321 1.6596029 1.52941294 0.80577843
## 300 -2.080897708 1.9151531169 -0.5224844 -0.29807769 -1.68851066
## 280 -1.292233482 0.1451564051 -0.5224844 -1.14062207 -1.64232012
## 13 -1.050660656 0.7863652700 -0.5224844 -0.57694801 -1.50374851
## 269 -0.567515004 -0.4383522101 -0.5224844 -0.85581834 -2.51994036
## 131 1.077311738 -0.7961887377 -0.6373311 0.17066179 1.26768382
## 299 -1.722091011 1.9151531169 -0.5224844 -0.29807769 -1.68851066
## 342 -0.684748875 1.5400302593 -0.9818712 -0.73715011 -1.36517689
## 9 1.116389695 1.0861216287 -0.5224844 -0.57694801 -1.50374851
## 69 -1.128816570 1.2836321952 -0.6373311 -0.37521203 0.20530143
## 370 1.002708366 -1.1579669285 1.6596029 1.52941294 0.80577843
## 19 -1.135921653 0.0006920764 -0.6373311 -0.60068166 1.17530274
## 487 0.398776300 -0.1183176566 1.6596029 1.52941294 0.80577843
## 278 -1.455650394 0.5070770657 -0.6373311 -0.91515245 -0.39517558
## 163 1.052443947 -0.8331358935 -0.5224844 -0.03107419 -1.73470120
## 390 1.077311738 -0.9815893753 1.6596029 1.52941294 0.80577843
## 413 1.116389695 -1.0643168114 1.6596029 1.52941294 0.80577843
## 414 1.116389695 -1.0474578907 1.6596029 1.52941294 0.80577843
## 195 -1.768274051 1.1514202651 -0.9818712 -0.84988492 -1.31898635
## 339 -1.111053862 0.4830471675 -0.5224844 -1.09315478 0.80577843
## 33 0.476932215 0.0925850666 -0.6373311 -0.60068166 1.17530274
## 273 -0.350809969 0.0581548764 -0.7521778 -1.09908819 0.06672981
## 475 0.952972784 -0.6483526248 1.6596029 1.52941294 0.80577843
## 470 -0.421860800 -0.4612898402 1.6596029 1.52941294 0.80577843
## 446 0.931657534 -0.8582105699 1.6596029 1.52941294 0.80577843
## 96 -0.382782843 -0.1423950448 -0.8670245 -0.78461740 -0.21041342
## 87 -0.833955621 0.3002109855 -0.7521778 -0.95668632 0.02053927
## 67 -1.331311439 1.3375332514 -0.6373311 -0.42267932 -1.08803366
## 417 0.789555872 -0.9381835908 1.6596029 1.52941294 0.80577843
## 99 -1.125264029 -0.1423950448 -0.8670245 -0.78461740 -0.21041342
## 324 0.203386515 0.4397838527 -0.5224844 -0.71934988 0.52863520
## 284 -1.555121557 0.9925190015 -0.9818712 -1.24742347 -2.24279713
## 174 0.551535588 -0.5455369536 -0.5224844 -0.66594918 -0.85708096
## 113 0.864159245 -0.6846349217 -0.4076377 0.14099473 -0.30279450
## 381 0.828633829 -1.1295679579 1.6596029 1.52941294 0.80577843
## 488 -0.546199754 -0.3052379716 1.6596029 1.52941294 0.80577843
## 343 -0.315284553 1.1738829960 -0.9818712 0.08166062 -1.18041474
## 14 -0.240681180 0.4333252240 -0.6373311 -0.60068166 1.17530274
## 224 0.434301716 -0.2483450505 -0.1779443 -0.60068166 -0.48755665
## 40 -1.661697804 0.7627152911 -0.7521778 -0.92701927 -0.07184181
## 372 1.116389695 -1.2470580136 1.6596029 1.52941294 0.80577843
## 316 0.324172928 0.0712146037 -0.6373311 -0.61848189 -0.02565127
## 422 0.949420242 -0.9120166463 1.6596029 1.52941294 0.80577843
## 466 -0.723826832 -0.3459843207 1.6596029 1.52941294 0.80577843
## 462 0.704294875 -0.5831489682 1.6596029 1.52941294 0.80577843
## 106 0.999155824 -0.8016975682 -0.5224844 -0.14380900 1.12911220
## 479 0.999155824 -0.7714939807 1.6596029 1.52941294 0.80577843
## 294 -1.782484217 0.8109650472 -0.6373311 -0.70748306 -1.13422420
## 115 0.555088129 -0.7306526517 -0.4076377 0.14099473 -0.30279450
## 125 0.967182950 -0.8494724251 -0.8670245 -1.30675758 0.29768250
## 416 1.116389695 -0.9309651233 1.6596029 1.52941294 0.80577843
## 364 0.725610124 -0.8977221812 1.6596029 1.52941294 0.80577843
## 261 0.469827132 -0.7992280924 -0.5224844 -0.85581834 -2.51994036
## 210 1.116389695 0.0379716616 -0.6373311 -0.77868399 0.06672981
## 331 -1.292233482 0.9871051509 -0.6373311 0.12912791 -0.71850935
## 385 0.803766038 -1.1186452769 1.6596029 1.52941294 0.80577843
## 411 1.116389695 -1.1312301050 1.6596029 1.52941294 0.80577843
## 84 -0.777114956 0.7625253315 -0.6373311 -0.75495035 0.25149196
## 177 -0.759352248 -0.1140435641 -0.5224844 -0.66594918 -0.85708096
## 225 0.345488177 -0.4277144686 -0.1779443 -0.60068166 -0.48755665
## 186 0.007996729 -0.2447358168 -0.7521778 -1.27709053 -0.30279450
## 256 -1.757616426 2.5764502168 -0.9818712 -0.55321437 -0.94946204
## 105 0.761135540 -0.6525317376 -0.5224844 -0.14380900 1.12911220
## 38 -0.961847117 0.0660856927 -0.5224844 -0.76681717 0.34387304
## 275 -1.267365691 0.1341862342 -0.6373311 -0.91515245 -0.39517558
## 441 0.846396537 -0.9160057994 1.6596029 1.52941294 0.80577843
## 22 0.732715207 0.1031753182 -0.6373311 -0.60068166 1.17530274
## 142 1.116389695 -1.1186927669 -0.6373311 0.17066179 1.26768382
## 7 -0.070159185 0.8384142195 -0.5224844 -0.57694801 -1.50374851
## 62 0.881921953 1.4358373805 -0.1779443 -0.73715011 0.57482574
## 423 0.675874542 -0.8756393696 1.6596029 1.52941294 0.80577843
## 266 -0.205155765 -0.8588754287 -0.5224844 -0.85581834 -2.51994036
## 75 -2.222999370 0.2167712006 -0.5224844 -0.06074124 0.11292035
## 397 0.974288033 -1.0059517029 1.6596029 1.52941294 0.80577843
## 79 -0.528437047 0.5789293108 -0.5224844 -0.06074124 0.11292035
## 129 1.073759197 -0.9415078850 -0.6373311 0.17066179 1.26768382
## 403 1.116389695 -1.0239028917 1.6596029 1.52941294 0.80577843
## 348 -1.452097852 2.2511442825 -0.6373311 -0.33961157 -0.25660396
## 245 0.281542429 1.9755128019 -0.2927910 -0.46421320 0.29768250
## 444 1.116389695 -0.8627221120 1.6596029 1.52941294 0.80577843
## 442 1.016918532 -0.8215483536 1.6596029 1.52941294 0.80577843
## 237 0.281542429 0.1676191361 -0.1779443 -0.60068166 -0.48755665
## 325 -1.011582699 0.4397838527 -0.5224844 -0.71934988 0.52863520
## 162 0.789555872 -0.8662838558 -0.5224844 -0.03107419 -1.73470120
## 435 0.938762618 -0.7469416934 1.6596029 1.52941294 0.80577843
## 144 1.116389695 -1.1317999841 -0.5224844 -0.03107419 -1.73470120
## 172 1.020471073 -0.6678709809 -0.5224844 -0.03107419 -1.73470120
## 336 -1.210525026 1.0401513886 -0.5224844 -1.09315478 0.80577843
## 160 1.116389695 -0.9636381865 -0.5224844 -0.03107419 -1.73470120
## 293 -1.604857139 0.6276539660 -0.6373311 -0.96855315 0.34387304
## 183 0.839291454 -0.5197499285 -0.7521778 -1.27709053 -0.30279450
## 83 -1.292233482 0.7625253315 -0.6373311 -0.75495035 0.25149196
## 132 1.034681240 -0.7237666137 -0.6373311 0.17066179 1.26768382
## 480 0.690084708 -0.8756393696 1.6596029 1.52941294 0.80577843
## 464 0.757582998 -0.4717851119 1.6596029 1.52941294 0.80577843
## 64 -0.894348827 1.9878601804 -0.1779443 -0.73715011 0.57482574
## 234 0.064837394 -0.0679783442 -0.1779443 -0.60068166 -0.48755665
## 264 0.920999910 -0.8150422349 -0.5224844 -0.85581834 -2.51994036
## 471 0.547983046 -0.3617034834 1.6596029 1.52941294 0.80577843
## 354 -1.153684361 3.9566021965 -0.5224844 -1.31269099 -0.67231881
## 137 0.885474494 -0.8681834525 -0.6373311 0.17066179 1.26768382
## 393 1.009813449 -0.9616910999 1.6596029 1.52941294 0.80577843
## 271 -0.940531867 0.3010658040 -0.7521778 -1.09908819 0.06672981
## 481 -0.137657475 -0.1761128861 1.6596029 1.52941294 0.80577843
## 168 0.377461051 -0.6502047316 -0.5224844 -0.03107419 -1.73470120
## 5 -0.510674339 1.0766711351 -0.7521778 -1.10502160 0.11292035
## 191 -1.672355429 1.2749890302 -0.5224844 -0.06074124 -1.50374851
## 383 1.116389695 -1.0517319833 1.6596029 1.52941294 0.80577843
## 92 0.189176348 -0.3338269018 -0.8670245 -0.82021787 -0.30279450
## 70 -1.263813149 1.2836321952 -0.6373311 -0.37521203 0.20530143
## 289 -0.816192913 1.6726695986 -0.4076377 -0.68374941 -0.85708096
## 272 -1.857087590 0.3010658040 -0.7521778 -1.09908819 0.06672981
## 161 0.853501620 -0.9482039634 -0.5224844 -0.03107419 -1.73470120
## 356 -1.743406260 3.2248775491 -0.6373311 -0.44047956 1.63720813
## 363 0.981393116 -0.8032647354 1.6596029 1.52941294 0.80577843
## 221 0.707847416 -0.4432436716 -0.1779443 -0.60068166 -0.48755665
## 247 -1.196314860 2.0232876588 -0.2927910 -0.46421320 0.29768250
## 415 1.116389695 -1.0147848276 1.6596029 1.52941294 0.80577843
## 445 0.995603282 -0.9020437636 1.6596029 1.52941294 0.80577843
## 25 0.906789743 0.2871037683 -0.6373311 -0.60068166 1.17530274
## 76 -0.837508162 0.3360183832 -0.5224844 -0.06074124 0.11292035
## 482 0.224701764 -0.2200410597 1.6596029 1.52941294 0.80577843
## 30 0.665216917 0.2108349609 -0.6373311 -0.60068166 1.17530274
## 277 -0.695406500 0.4711746882 -0.6373311 -0.91515245 -0.39517558
## 361 0.690084708 -0.6063715378 1.6596029 1.52941294 0.80577843
## 48 0.601271169 0.8996287230 -0.7521778 -1.03975408 -0.25660396
## 338 -0.318837095 0.8642962245 -0.5224844 -1.09315478 0.80577843
## 128 0.974288033 -0.9530004450 -0.6373311 0.17066179 1.26768382
## 133 1.041786323 -0.6969823003 -0.6373311 0.17066179 1.26768382
## 253 -2.194579038 2.4275218358 -0.2927910 -0.46421320 0.29768250
## 73 -2.159053622 0.7086717651 -0.6373311 -0.61254848 0.34387304
## 156 0.498247464 -0.9733261297 -0.5224844 -0.03107419 -1.73470120
## 365 0.508905089 -0.8977221812 1.6596029 1.52941294 0.80577843
## 312 -0.560409921 -0.5483863487 -0.6373311 -0.61848189 -0.02565127
## 355 -1.658145262 3.2248775491 -0.6373311 -0.44047956 1.63720813
## 504 0.796660955 -0.7729186782 -0.9818712 -0.80241764 1.17530274
## 3 -0.265548971 0.5566090496 -0.8670245 -0.98635338 -0.30279450
## 192 -1.341969064 1.2749890302 -0.5224844 -0.06074124 -1.50374851
## 402 1.116389695 -1.0547238481 1.6596029 1.52941294 0.80577843
## 109 1.013365990 -0.6468804374 -0.5224844 -0.14380900 1.12911220
## 55 -0.745142082 1.6738568465 -0.7521778 0.36053095 1.22149328
## 119 0.160756016 -0.6256999342 -0.4076377 0.14099473 -0.30279450
## 489 0.857054162 -0.9375187319 -0.6373311 1.79641644 0.75958789
## 139 1.052443947 -1.0098458762 -0.6373311 0.17066179 1.26768382
## 185 0.754030456 -0.3833113958 -0.7521778 -1.27709053 -0.30279450
## 499 -0.116342226 -0.6581830377 -0.4076377 -0.10227512 0.34387304
## 58 -0.997372532 2.1511780064 -0.5224844 -0.90328562 -1.54993904
## 463 0.512457630 -0.5036983364 1.6596029 1.52941294 0.80577843
## 407 1.116389695 -1.2427839210 1.6596029 1.52941294 0.80577843
## 352 -1.160789444 3.2840499862 -0.6373311 0.01639310 -0.07184181
## 302 -1.000925074 0.8057411563 -0.2927910 -0.47014661 -1.08803366
## 41 -1.874850298 0.7627152911 -0.7521778 -0.92701927 -0.07184181
## 357 1.024023615 -0.7944316108 1.6596029 1.52941294 0.80577843
## 114 0.952972784 -0.5922195425 -0.4076377 0.14099473 -0.30279450
## 401 1.116389695 -1.0477428302 1.6596029 1.52941294 0.80577843
## 420 0.281542429 -0.9502935197 1.6596029 1.52941294 0.80577843
## 484 -1.004477616 0.1440166471 1.6596029 1.52941294 0.80577843
## 153 0.690084708 -1.0375799879 -0.5224844 -0.03107419 -1.73470120
## 386 1.048891406 -1.1250089259 1.6596029 1.52941294 0.80577843
## 287 -1.317101273 2.5141909351 -0.9818712 -0.99228679 -0.11803234
## 493 0.530220338 -0.8002728706 -0.6373311 1.79641644 0.75958789
## 369 1.116389695 -1.1694594886 1.6596029 1.52941294 0.80577843
## 29 0.917447368 0.3132707128 -0.6373311 -0.60068166 1.17530274
## 42 -2.333128159 0.9145880470 -0.7521778 -1.03975408 -0.25660396
## 108 0.590613545 -0.7943366310 -0.5224844 -0.14380900 1.12911220
## 11 0.913894827 1.2117799501 -0.5224844 -0.57694801 -1.50374851
## 24 1.116389695 0.1425444597 -0.6373311 -0.60068166 1.17530274
## 206 -1.643935096 0.0714045634 -0.6373311 -0.77868399 0.06672981
## 360 0.452064424 -0.6106931203 1.6596029 1.52941294 0.80577843
## 18 0.466274590 0.2198105553 -0.6373311 -0.60068166 1.17530274
## 81 -1.246050442 0.7625253315 -0.6373311 -0.75495035 0.25149196
## 432 0.913894827 -0.8105781827 1.6596029 1.52941294 0.80577843
## 315 0.665216917 -0.0915333432 -0.6373311 -0.61848189 -0.02565127
## 297 -0.624355669 1.0283263992 -0.6373311 -0.70748306 -1.13422420
## 71 -2.201684121 0.7086717651 -0.6373311 -0.61254848 0.34387304
## 170 0.945867701 -0.7278032567 -0.5224844 -0.03107419 -1.73470120
## 150 0.935210076 -1.0777089681 -0.5224844 -0.03107419 -1.73470120
## 404 0.974288033 -0.9936043244 1.6596029 1.52941294 0.80577843
## 303 -1.782484217 0.8057411563 -0.2927910 -0.47014661 -1.08803366
## 238 0.107467893 0.1676191361 -0.1779443 -0.60068166 -0.48755665
## 318 0.111020434 0.1123883621 -0.6373311 -0.61848189 -0.02565127
## 47 -1.253155525 0.6199131095 -0.7521778 -1.03975408 -0.25660396
## 187 -0.531989588 -0.2829652003 -0.7521778 -1.27709053 -0.30279450
## 17 -1.395257187 0.3341187865 -0.6373311 -0.60068166 1.17530274
## 304 -1.807352008 0.8057411563 -0.2927910 -0.47014661 -1.08803366
## 20 0.032864520 0.0006920764 -0.6373311 -0.60068166 1.17530274
## 154 1.063101572 -1.0314062987 -0.5224844 -0.03107419 -1.73470120
## 68 -1.675907970 1.2836321952 -0.6373311 -0.37521203 0.20530143
## 461 0.761135540 -0.5687120333 1.6596029 1.52941294 0.80577843
## 424 0.587061003 -0.8421114879 1.6596029 1.52941294 0.80577843
## 451 0.853501620 -0.6987869171 1.6596029 1.52941294 0.80577843
## 257 -1.221182651 1.2067460189 -0.7521778 -0.97448656 -1.18041474
## 501 0.395223759 -0.6158695213 -0.4076377 -0.10227512 0.34387304
## 503 0.288647512 -0.7159307773 -0.9818712 -0.80241764 1.17530274
## 396 1.073759197 -0.9827291333 1.6596029 1.52941294 0.80577843
## 211 0.835738912 0.0389689498 -0.6373311 -0.77868399 0.06672981
## 126 0.704294875 -0.8558360740 -0.8670245 -1.30675758 0.29768250
## 368 1.116389695 -1.0848799457 1.6596029 1.52941294 0.80577843
## 384 1.116389695 -1.0741947142 1.6596029 1.52941294 0.80577843
## 1 -0.119894767 0.1400749840 -0.9818712 -0.66594918 -1.45755797
## 335 -1.068423364 1.2648261879 -0.5224844 -1.09315478 0.80577843
## 196 -1.299338565 0.8801578569 -0.6373311 -0.90921904 -1.87327282
## 44 -2.205236663 0.9145880470 -0.7521778 -1.03975408 -0.25660396
## 286 -1.302891107 1.6679680968 -0.9818712 -0.64221553 -1.45755797
## 391 1.009813449 -0.8873693792 1.6596029 1.52941294 0.80577843
## 203 -1.878402839 1.1753551835 -0.8670245 -0.35741180 -1.73470120
## 491 1.045338864 -0.9367114033 -0.6373311 1.79641644 0.75958789
## 421 1.116389695 -0.9194725633 1.6596029 1.52941294 0.80577843
## 31 0.906789743 0.2079855659 -0.6373311 -0.60068166 1.17530274
## 167 0.981393116 -0.8306664178 -0.5224844 -0.03107419 -1.73470120
## 274 -0.595935336 0.2713846056 -0.7521778 -1.09908819 0.06672981
## 497 0.153650933 -0.4732098094 -0.4076377 -0.10227512 0.34387304
## 179 0.206939056 -0.4177890758 -0.5224844 -0.66594918 -0.85708096
## 250 -1.814457091 1.9145357480 -0.2927910 -0.46421320 0.29768250
## 181 0.523115255 -0.5005640019 -0.7521778 -1.27709053 -0.30279450
## 243 -0.556857379 1.5388905012 -0.4076377 -0.64221553 -0.85708096
## 469 0.086152643 -0.4210658801 1.6596029 1.52941294 0.80577843
## 419 1.116389695 -0.9462093868 1.6596029 1.52941294 0.80577843
## 217 -0.446728591 -0.3243289184 -0.5224844 -0.78461740 -0.94946204
## 251 -1.974321461 1.7104240829 -0.2927910 -0.46421320 0.29768250
## 436 0.924552451 -0.7932443629 1.6596029 1.52941294 0.80577843
## 136 1.052443947 -0.7998929513 -0.6373311 0.17066179 1.26768382
## 447 0.988498199 -0.8182715493 1.6596029 1.52941294 0.80577843
## 376 1.041786323 -1.1771528552 1.6596029 1.52941294 0.80577843
## 169 0.977840575 -0.8049743725 -0.5224844 -0.03107419 -1.73470120
## 320 -0.347257427 0.0982363667 -0.6373311 -0.61848189 -0.02565127
## 428 0.359698343 -0.9175729666 1.6596029 1.52941294 0.80577843
## 117 0.139440767 -0.5057404029 -0.4076377 0.14099473 -0.30279450
## 495 -0.922769160 -0.6711952751 -0.4076377 -0.10227512 0.34387304
## 483 0.299305137 -0.1825715149 1.6596029 1.52941294 0.80577843
## 212 0.711399958 -0.0617571650 -0.6373311 -0.77868399 0.06672981
## 313 0.775345706 -0.4563983787 -0.6373311 -0.61848189 -0.02565127
## 205 -1.302891107 0.6282713349 -0.6373311 -1.09315478 -1.73470120
## 86 -0.443176049 0.3051974268 -0.7521778 -0.95668632 0.02053927
## 151 1.020471073 -1.0338757744 -0.5224844 -0.03107419 -1.73470120
## 282 -1.114606404 0.6884410603 -0.5224844 -1.14062207 -1.64232012
## 26 0.608376252 0.3132232229 -0.6373311 -0.60068166 1.17530274
## 209 -0.336599802 0.2109299408 -0.6373311 -0.77868399 0.06672981
## 103 0.597718628 -0.5130538501 -0.5224844 -0.14380900 1.12911220
## 433 0.221149222 -0.7572944954 1.6596029 1.52941294 0.80577843
## 135 1.059549031 -0.6881491756 -0.6373311 0.17066179 1.26768382
## 506 0.434301716 -0.6126402069 -0.9818712 -0.80241764 1.17530274
## 373 0.746925373 -1.2658165310 1.6596029 1.52941294 0.80577843
## 310 0.288647512 -0.3288879504 -0.6373311 -0.61848189 -0.02565127
## 241 -0.507121797 1.2067460189 -0.4076377 -0.64221553 -0.85708096
## 122 0.551535588 -0.7587191929 -0.8670245 -1.30675758 0.29768250
## 182 -0.226471014 -0.5685220737 -0.7521778 -1.27709053 -0.30279450
## 412 1.116389695 -1.0768541496 1.6596029 1.52941294 0.80577843
## 149 0.896132119 -1.0758568614 -0.5224844 -0.03107419 -1.73470120
## 486 -0.592382795 0.0933923952 1.6596029 1.52941294 0.80577843
## 219 0.896132119 -0.4301364543 -0.5224844 -0.78461740 -0.94946204
## 262 0.739820290 -0.7860733853 -0.5224844 -0.85581834 -2.51994036
## 123 0.864159245 -0.8111955516 -0.8670245 -1.30675758 0.29768250
## 474 -0.034633770 -0.5993905200 1.6596029 1.52941294 0.80577843
## 98 0.263779721 -0.1423950448 -0.8670245 -0.78461740 -0.21041342
## 120 -0.119894767 -0.4919208369 -0.4076377 0.14099473 -0.30279450
## 82 0.064837394 0.7625253315 -0.6373311 -0.75495035 0.25149196
## 198 -1.135921653 1.6687754254 -0.8670245 -0.47014661 -2.70470251
## 308 0.061284852 -0.2908010367 -0.2927910 -1.10502160 -0.02565127
## 218 0.587061003 -0.1775850736 -0.5224844 -0.78461740 -0.94946204
## 197 -1.224735192 1.6687754254 -0.8670245 -0.47014661 -2.70470251
## 32 1.116389695 0.1804414138 -0.6373311 -0.60068166 1.17530274
## 505 0.736267749 -0.6677760011 -0.9818712 -0.80241764 1.17530274
## 175 0.004444187 -0.5191325596 -0.5224844 -0.66594918 -0.85708096
## 455 0.906789743 -0.6168668096 1.6596029 1.52941294 0.80577843
## 226 0.512457630 -0.4277144686 -0.1779443 -0.60068166 -0.48755665
## 319 -0.048843936 -0.1246813056 -0.6373311 -0.61848189 -0.02565127
## 12 0.508905089 1.1547920492 -0.5224844 -0.57694801 -1.50374851
## 23 0.821528746 0.0863638874 -0.6373311 -0.60068166 1.17530274
## 213 -0.524884505 -0.0676459148 -0.6373311 -0.77868399 0.06672981
## 4 -0.809087830 1.0766711351 -0.7521778 -1.10502160 0.11292035
## 61 -0.084369352 1.6290738544 -0.1779443 -0.73715011 0.57482574
## 223 0.324172928 -0.2483450505 -0.1779443 -0.60068166 -0.48755665
## 45 -1.015135240 0.9145880470 -0.7521778 -1.03975408 -0.25660396
## 434 0.686532167 -0.7024911307 1.6596029 1.52941294 0.80577843
## 152 1.116389695 -1.0464131126 -0.5224844 -0.03107419 -1.73470120
## 258 0.651006751 -0.9469692255 -0.5224844 -0.85581834 -2.51994036
## 358 0.796660955 -0.6125452271 1.6596029 1.52941294 0.80577843
## 222 0.807318580 -0.3547699554 -0.1779443 -0.60068166 -0.48755665
## 472 0.786003330 -0.3304076278 1.6596029 1.52941294 0.80577843
## 260 1.116389695 -0.8473828687 -0.5224844 -0.85581834 -2.51994036
## 448 0.995603282 -0.7584342534 1.6596029 1.52941294 0.80577843
## 490 1.055996489 -0.9686246278 -0.6373311 1.79641644 0.75958789
## 341 -0.357915052 0.4830471675 -0.5224844 -1.09315478 0.80577843
## 458 0.416539008 -0.4824228534 1.6596029 1.52941294 0.80577843
## 220 0.846396537 -0.2050342458 -0.5224844 -0.78461740 -0.94946204
## 426 0.952972784 -0.8953951752 1.6596029 1.52941294 0.80577843
## 399 1.116389695 -1.0948528283 1.6596029 1.52941294 0.80577843
## 459 0.537325421 -0.4805707466 1.6596029 1.52941294 0.80577843
## 306 -0.372125218 -0.2018524214 -0.2927910 -1.10502160 -0.02565127
## 157 0.903237202 -0.9776477121 -0.5224844 -0.03107419 -1.73470120
## 366 0.686532167 -1.0361552904 1.6596029 1.52941294 0.80577843
## 216 -0.929874243 0.0714045634 -0.6373311 -0.77868399 0.06672981
## 102 0.096810268 -0.4459031069 -0.5224844 -0.14380900 1.12911220
## 54 -1.675907970 1.4340327636 -0.6373311 -0.98041997 -0.76469989
## 190 -1.054213197 0.3664594203 -0.5224844 -0.06074124 -1.50374851
## 410 1.116389695 -1.1062978984 1.6596029 1.52941294 0.80577843
## 193 -1.501833434 1.2749890302 -0.5224844 -0.06074124 -1.50374851
## 240 -0.936979326 1.1373157596 -0.4076377 -0.64221553 -0.85708096
## 473 0.228254306 -0.4267171803 1.6596029 1.52941294 0.80577843
## 452 1.052443947 -0.6837801032 1.6596029 1.52941294 0.80577843
## 27 0.771793164 0.4212152950 -0.6373311 -0.60068166 1.17530274
## 80 -1.135921653 0.3360183832 -0.5224844 -0.06074124 0.11292035
## 485 -0.947636951 -0.0337381137 1.6596029 1.52941294 0.80577843
## 166 0.867711786 -0.7178778639 -0.5224844 -0.03107419 -1.73470120
## 8 0.977840575 1.0236248974 -0.5224844 -0.57694801 -1.50374851
## 178 0.171413641 -0.2267846280 -0.5224844 -0.66594918 -0.85708096
## 39 -1.363284313 0.0248169544 -0.5224844 -0.76681717 0.34387304
## 232 0.295752595 -0.0586703204 -0.1779443 -0.60068166 -0.48755665
## 408 1.116389695 -1.1919222195 1.6596029 1.52941294 0.80577843
## 165 0.825081288 -0.6520568384 -0.5224844 -0.03107419 -1.73470120
## 438 1.116389695 -0.8932106390 1.6596029 1.52941294 0.80577843
## 281 -0.144762558 0.4272465145 -0.5224844 -1.14062207 -1.64232012
## 37 -0.254891346 -0.1981007179 -0.5224844 -0.76681717 0.34387304
## 345 -1.437887686 1.2681504821 -0.5224844 -0.22687676 -0.39517558
## 439 0.686532167 -0.9376612017 1.6596029 1.52941294 0.80577843
## 477 0.889027036 -0.7074775720 1.6596029 1.52941294 0.80577843
## 141 0.889027036 -1.0367726593 -0.6373311 0.17066179 1.26768382
## 124 1.009813449 -0.8788686839 -0.8670245 -1.30675758 0.29768250
## 201 -1.942348587 1.8323307009 -0.7521778 -0.03700760 -0.67231881
## 35 1.006260907 -0.0167367233 -0.6373311 -0.60068166 1.17530274
## 173 0.707847416 -0.5693768922 -0.5224844 -0.66594918 -0.85708096
## 233 0.167861099 0.0205903518 -0.1779443 -0.60068166 -0.48755665
## 465 -0.112789684 -0.3949464255 1.6596029 1.52941294 0.80577843
## 496 -1.413019895 -0.4732098094 -0.4076377 -0.10227512 0.34387304
## 10 0.615481336 1.3283202075 -0.5224844 -0.57694801 -1.50374851
## 460 0.562193212 -0.5117241325 1.6596029 1.52941294 0.80577843
## 121 0.039969603 -0.7300827727 -0.8670245 -1.30675758 0.29768250
## 94 -1.409467353 -0.0613297558 -0.6373311 -0.82021787 -0.11803234
## 56 -1.658145262 2.3277455193 -0.5224844 -1.08128796 -0.25660396
## 50 -0.233576097 1.0887810641 -0.7521778 -1.03975408 -0.25660396
## 350 -1.210525026 2.3730983904 -0.9818712 -0.43454615 0.57482574
## 449 1.070206655 -0.7282306659 1.6596029 1.52941294 0.80577843
## 307 0.118125517 -0.3304551177 -0.2927910 -1.10502160 -0.02565127
## 143 1.116389695 -1.1746358896 -0.5224844 -0.03107419 -1.73470120
## 130 0.928104993 -0.8620097633 -0.6373311 0.17066179 1.26768382
## 21 1.048891406 0.0013569352 -0.6373311 -0.60068166 1.17530274
## 140 1.041786323 -1.0097983862 -0.6373311 0.17066179 1.26768382
## 263 0.814423663 -0.7154558781 -0.5224844 -0.85581834 -2.51994036
## 107 0.828633829 -0.7522605641 -0.5224844 -0.14380900 1.12911220
## 244 -2.159053622 1.5388905012 -0.4076377 -0.64221553 -0.85708096
## 425 0.071942477 -0.8223081923 1.6596029 1.52941294 0.80577843
## 252 -2.119975665 1.7104240829 -0.2927910 -0.46421320 0.29768250
## 85 -0.730931915 0.4674704746 -0.7521778 -0.95668632 0.02053927
## 248 0.377461051 2.0232876588 -0.2927910 -0.46421320 0.29768250
## 246 0.057732311 1.9755128019 -0.2927910 -0.46421320 0.29768250
## 59 -1.398809729 1.9089794276 -0.1779443 -0.73715011 0.57482574
## 322 -0.507121797 0.3539695720 -0.5224844 -0.71934988 0.52863520
## 227 0.636796585 -0.2751293639 -0.1779443 -0.60068166 -0.48755665
## 340 -0.823297996 0.4830471675 -0.5224844 -1.09315478 0.80577843
## 326 -1.913928255 0.7697437989 -0.5224844 -0.71934988 0.52863520
## 377 0.878369411 -1.1635707388 1.6596029 1.52941294 0.80577843
## 346 -0.713169208 2.0033893834 -0.7521778 -0.33367816 0.15911089
## 301 -0.752247165 1.9151531169 -0.5224844 -0.29807769 -1.68851066
## 291 -1.444992769 0.6276539660 -0.6373311 -0.96855315 0.34387304
## 440 0.899684660 -0.9392758589 1.6596029 1.52941294 0.80577843
## 208 0.146545850 0.2658757752 -0.6373311 -0.77868399 0.06672981
## 89 0.629691502 -0.1773001341 -0.8670245 -0.82021787 -0.30279450
## 380 1.116389695 -1.1440048928 1.6596029 1.52941294 0.80577843
## 268 -0.055949019 -0.6522467981 -0.5224844 -0.85581834 -2.51994036
## 127 0.960077867 -0.9677698093 -0.8670245 -1.30675758 0.29768250
## 194 -2.084450250 1.1514202651 -0.9818712 -0.84988492 -1.31898635
## 148 0.963630408 -1.1085299245 -0.5224844 -0.03107419 -1.73470120
## 104 0.668769459 -0.5130538501 -0.5224844 -0.14380900 1.12911220
## 349 -1.381047021 2.1602960705 -0.6373311 -0.76088376 -0.67231881
## 429 0.338383094 -0.8830477967 1.6596029 1.52941294 0.80577843
## 57 -1.167894527 2.5609210138 -0.8670245 -0.56508119 -0.53374719
## 409 1.041786323 -1.1114268095 1.6596029 1.52941294 0.80577843
## 321 -0.578172628 0.3539695720 -0.5224844 -0.71934988 0.52863520
## 467 0.572850837 -0.4385896596 1.6596029 1.52941294 0.80577843
## 398 1.077311738 -1.0265623271 1.6596029 1.52941294 0.80577843
## 468 0.920999910 -0.5958762661 1.6596029 1.52941294 0.80577843
## 134 0.952972784 -0.6293091680 -0.6373311 0.17066179 1.26768382
## 270 -0.251338805 0.0581548764 -0.7521778 -1.09908819 0.06672981
## 406 1.116389695 -1.1253413553 1.6596029 1.52941294 0.80577843
## 305 -0.976057283 0.1077818401 -0.2927910 -1.10502160 -0.02565127
## 111 -0.503569256 -0.4830877123 -0.5224844 -0.14380900 1.12911220
## 389 1.116389695 -1.0474104008 1.6596029 1.52941294 0.80577843
## black lstat medv crime
## 353 0.390558315 -0.681000049 -0.427613550 low
## 176 0.375551994 -1.025486649 0.746669127 low
## 283 0.223407620 -1.350368321 2.551585092 low
## 100 0.440615895 -0.905056374 1.159842661 low
## 359 0.424514223 -0.164270149 0.018178948 high
## 337 0.440615895 -0.399529291 -0.329756660 low
## 53 0.440615895 -1.032488409 0.268257666 low
## 431 -2.992764527 0.698346703 -0.873406047 high
## 430 -3.241738006 1.600173411 -1.417055434 high
## 180 0.440615895 -1.066096858 1.594762170 low
## 323 0.440615895 -0.693603218 -0.231899770 med_high
## 60 0.440615895 -0.480749709 -0.318883672 med_low
## 15 0.255720500 -0.335113097 -0.471105500 med_high
## 95 0.440615895 -0.288901480 -0.210153795 low
## 314 0.402169045 -0.665596177 -0.101423917 med_high
## 74 0.224502972 -0.716008850 0.094289862 med_low
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## 439 -3.151590547 2.992123331 -1.536658299 high
## 477 0.433057967 0.843983314 -0.634200317 high
## 141 0.344005860 1.611376228 -0.927770986 med_high
## 124 0.149361834 1.786420232 -0.568962390 med_low
## 201 0.302601560 -1.148717628 1.127223698 low
## 35 -1.186967442 1.076441751 -0.982135924 med_high
## 173 0.440615895 0.285242853 0.061670899 med_low
## 233 0.320236725 -1.425987330 2.084046619 med_high
## 465 0.440615895 0.079391104 -0.123169893 high
## 496 0.401073693 0.692745294 0.061670899 med_low
## 10 0.328999540 0.622727693 -0.394994586 med_low
## 460 0.440615895 0.286643205 -0.275391721 high
## 121 0.355726125 0.240431588 -0.057931966 low
## 94 0.434372389 -0.902255670 0.268257666 low
## 56 0.429990982 -1.098304955 1.399048391 low
## 50 0.440615895 0.496696010 -0.340629648 med_low
## 350 0.363393588 -0.947066935 0.442225470 low
## 449 0.440615895 0.766963952 -0.916897998 high
## 307 0.440615895 -0.865846518 1.181588636 low
## 143 0.440615895 1.983869868 -0.993008912 med_high
## 130 0.440615895 0.796371345 -0.895152022 med_high
## 21 0.217930861 1.171665689 -0.971262937 med_high
## 140 0.440615895 0.813175569 -0.514597451 med_high
## 263 0.330642568 -0.944266231 2.856028748 med_high
## 107 0.427143067 0.841182610 -0.329756660 med_low
## 244 0.197557316 -1.045091578 0.126908825 med_low
## 425 -3.866855315 0.631129805 -1.177849704 high
## 252 0.223407620 -1.269147903 0.246511690 med_low
## 85 0.440615895 -0.424735627 0.148654801 low
## 248 0.213220848 -0.350516970 -0.221026782 med_low
## 246 0.355507055 0.813175569 -0.438486537 med_low
## 59 0.372485009 -0.811232788 0.083416874 med_low
## 322 0.440615895 -0.809832436 0.061670899 med_low
## 227 0.336338397 -1.333564096 1.638254121 med_high
## 340 0.440615895 -0.407931403 -0.384121599 low
## 326 0.405345565 -1.060495450 0.224765715 med_low
## 377 0.069510683 1.482543841 -0.938643973 high
## 346 0.317279275 -0.297303592 -0.547216415 low
## 301 0.374456642 -0.921860599 0.246511690 low
## 291 0.440615895 -1.305557056 0.648812237 low
## 440 0.440615895 1.432131167 -1.058246839 high
## 208 0.358793111 0.757161488 -0.003567028 med_low
## 89 0.440615895 -1.001680665 0.116035838 low
## 380 0.406002776 1.278092444 -1.340944520 high
## 268 0.305230404 -0.730012370 2.986504601 med_high
## 127 0.028654058 2.045485358 -0.742930194 med_high
## 194 0.401949974 -1.067497210 0.931509918 low
## 148 0.383767133 2.363365269 -0.862533059 med_high
## 104 0.413998845 0.110198849 -0.351502635 med_low
## 349 0.375332924 -0.933063415 0.213892727 low
## 429 -2.847301799 1.241683291 -1.253960618 high
## 57 0.440615895 -0.963871160 0.235638703 low
## 409 -0.460420549 1.925055083 -0.579835378 high
## 321 0.440615895 -0.763620819 0.137781813 med_low
## 467 -3.665748713 0.629729453 -0.384121599 high
## 398 0.398992524 1.017626966 -1.525785311 high
## 468 -0.278044464 1.213676250 -0.373248611 high
## 134 0.350687507 0.332854822 -0.449359525 med_high
## 270 0.379714331 0.139606242 -0.199280807 med_low
## 406 0.309940417 1.446134688 -1.906339882 high
## 305 0.405345565 -0.801430324 1.475159305 low
## 111 0.403264396 0.048583360 -0.090550930 med_low
## 389 0.177950518 2.516003640 -1.340944520 high
## zn indus chas nox rm age
## 6 -0.4872402 -1.30558569 -0.2723291 -0.83445805 0.206891639 -0.35080997
## 16 -0.4872402 -0.43682573 -0.2723291 -0.14407485 -0.641365488 -0.42896588
## 28 -0.4872402 -0.43682573 -0.2723291 -0.14407485 -0.338213193 0.71850504
## 34 -0.4872402 -0.43682573 -0.2723291 -0.14407485 -0.830657767 0.93876262
## 36 -0.4872402 -0.75459363 -0.2723291 -0.48063666 -0.500463717 -0.01331852
## 43 -0.4872402 -0.61611679 -0.2723291 -0.92075595 -0.164576667 -2.20168412
## 46 -0.4872402 -0.61611679 -0.2723291 -0.92075595 -0.857699521 -1.23539282
## 49 -0.4872402 -0.61611679 -0.2723291 -0.92075595 -1.260479332 0.94942024
## 51 0.4131797 -0.80123846 -0.2723291 -0.99842406 -0.457766211 -0.81264037
## 52 0.4131797 -0.80123846 -0.2723291 -0.99842406 -0.241432178 -0.19805068
## 63 0.5846882 -0.87557866 -0.2723291 -0.87760700 0.243896145 -0.02752869
## 65 0.2631097 -1.42219777 -0.2723291 -1.19604625 1.166162284 -0.32238964
## 66 2.9429307 -1.13212523 -0.2723291 -1.35224545 0.007636609 -1.80379947
## 77 -0.4872402 0.24681257 -0.2723291 -1.01568364 -0.008019143 0.21049160
## 78 -0.4872402 0.24681257 -0.2723291 -1.01568364 -0.205850923 -0.80908783
## 88 -0.4872402 -0.96886832 -0.2723291 -0.91212616 -0.232892677 -0.41830826
## 90 -0.4872402 -1.12629463 -0.2723291 -0.56693456 1.130581029 -0.19449814
## 91 -0.4872402 -1.12629463 -0.2723291 -0.56693456 0.188389387 -0.08792189
## 93 0.7133196 0.56895343 -0.2723291 -0.78267931 0.223970642 -0.53198959
## 97 -0.4872402 -1.20209248 -0.2723291 -0.94664532 -0.173116168 0.03641706
## 101 -0.4872402 -0.37560439 -0.2723291 -0.29941107 0.629596953 0.40232884
## 110 -0.4872402 -0.37560439 -0.2723291 -0.29941107 -0.079181654 0.80376604
## 112 -0.4872402 -0.16424500 -0.2723291 -0.06640675 0.612517950 0.46272205
## 116 -0.4872402 -0.16424500 -0.2723291 -0.06640675 -0.507579968 0.69718979
## 118 -0.4872402 -0.16424500 -0.2723291 -0.06640675 -0.375217698 0.49824746
## 138 -0.4872402 1.56744433 -0.2723291 0.59808708 0.241049645 1.05954903
## 145 -0.4872402 1.23072696 -0.2723291 2.72964520 -1.966411438 1.03823378
## 147 -0.4872402 1.23072696 -0.2723291 2.72964520 -0.934555033 1.11638970
## 155 -0.4872402 1.23072696 3.6647712 2.72964520 -0.221506675 0.97428803
## 158 -0.4872402 1.23072696 -0.2723291 0.43412107 0.937018999 1.02402361
## 159 -0.4872402 1.23072696 -0.2723291 0.43412107 -0.311171439 1.11638970
## 171 -0.4872402 1.23072696 -0.2723291 0.43412107 -0.583012230 0.92455245
## 184 -0.4872402 -1.26477147 -0.2723291 -0.57556435 0.396183918 0.96007787
## 188 1.4422310 -1.12192167 -0.2723291 -1.01568364 0.707875715 -0.97605728
## 189 1.4422310 -1.12192167 -0.2723291 -1.01568364 0.386221166 -1.40236227
## 199 2.9429307 -1.40179066 -0.2723291 -1.30046671 1.408114820 -1.07552845
## 200 3.5860878 -1.40907891 -0.2723291 -1.30909650 0.982563006 -1.89261301
## 202 3.0501236 -1.32745046 -0.2723291 -1.20553902 -0.174539418 -1.07197591
## 204 3.5860878 -1.23270315 -0.2723291 -1.19604625 2.232176694 -1.25670807
## 207 -0.4872402 -0.07970124 -0.2723291 -0.56693456 0.058873617 -0.57106755
## 214 -0.4872402 -0.07970124 -0.2723291 -0.56693456 0.128612878 -1.28868094
## 215 -0.4872402 -0.07970124 -0.2723291 -0.56693456 -1.241977079 -2.08800279
## 228 -0.4872402 -0.71961001 -0.2723291 -0.43748771 1.250134047 0.40232884
## 229 -0.4872402 -0.71961001 -0.2723291 -0.43748771 1.994493909 -1.83221980
## 230 -0.4872402 -0.71961001 -0.2723291 -0.43748771 0.380528166 -1.67590797
## 231 -0.4872402 -0.71961001 -0.2723291 -0.43748771 -0.432147707 -0.01687106
## 235 -0.4872402 -0.71961001 3.6647712 -0.41159834 0.628173703 -0.07371173
## 236 -0.4872402 -0.71961001 -0.2723291 -0.41159834 -0.282706434 -0.25133880
## 239 0.7990739 -0.90473168 -0.2723291 -1.09335175 0.279477400 -1.77893168
## 242 0.7990739 -0.90473168 -0.2723291 -1.09335175 -0.269897182 -0.12344731
## 249 0.4560568 -0.76917014 -0.2723291 -1.06746238 0.211161390 -0.69185396
## 254 0.4560568 -0.76917014 -0.2723291 -1.06746238 2.810016281 -2.13773837
## 255 2.9429307 -1.09276866 -0.2723291 -1.40402419 -0.251394930 -1.29933856
## 265 0.3703025 -1.04466617 -0.2723291 0.79657225 1.311333806 0.81797620
## 267 0.3703025 -1.04466617 -0.2723291 0.79657225 1.038069765 0.56929830
## 276 1.2278453 -0.68899934 -0.2723291 -0.92938574 0.810349730 -0.91566408
## 279 1.2278453 -0.68899934 -0.2723291 -0.92938574 0.280900651 -1.29578602
## 288 1.7638095 -0.84788329 -0.2723291 -1.29183692 -0.107646658 -1.32420636
## 290 1.7638095 -0.84788329 -0.2723291 -1.29183692 0.399030418 -1.62261985
## 292 2.9429307 -0.90181638 -0.2723291 -1.24005818 1.228785293 -1.45209785
## 295 -0.4872402 0.40569652 -0.2723291 -1.01568364 -0.392296701 -0.93342678
## 298 -0.4872402 0.40569652 -0.2723291 -1.01568364 -0.703988498 -0.37567776
## 309 -0.4872402 -0.18027916 -0.2723291 -0.09229612 0.498657933 0.49469492
## 311 -0.4872402 -0.18027916 -0.2723291 -0.09229612 -1.866783923 -1.09329115
## 317 -0.4872402 -0.18027916 -0.2723291 -0.09229612 -0.527505471 0.51956271
## 327 -0.4872402 -0.54760720 -0.2723291 -0.53241540 0.038948114 -1.40946735
## 328 -0.4872402 -0.54760720 -0.2723291 -0.53241540 -0.286976185 -0.88369120
## 329 -0.4872402 -1.15107469 -0.2723291 -0.81719847 -0.592974981 -1.51959614
## 330 -0.4872402 -1.15107469 -0.2723291 -0.81719847 0.068836369 -1.82511472
## 332 1.0134596 -0.74001712 -0.2723291 -1.00791683 -0.823541516 -1.42723006
## 333 1.0134596 -0.74001712 -0.2723291 -1.00791683 -0.360985196 -1.60840968
## 334 -0.4872402 -0.86683276 -0.2723291 -0.34256002 0.044641115 -1.08263353
## 344 1.8710023 -1.07236154 -0.2723291 -0.61008351 0.585476196 -0.43251842
## 347 -0.4872402 -0.98344483 -0.2723291 -0.97253469 -0.550277475 -0.57817263
## 351 1.2278453 -1.44114723 -0.2723291 -1.08472196 0.292286652 -0.85882341
## 362 -0.4872402 1.01499462 -0.2723291 1.85803641 -0.047870149 0.80021350
## 371 -0.4872402 1.01499462 3.6647712 0.65849561 1.040916265 1.02757616
## 374 -0.4872402 1.01499462 -0.2723291 0.97779784 -1.962141687 1.11638970
## 375 -0.4872402 1.01499462 -0.2723291 0.97779784 -3.055197852 1.11638970
## 378 -0.4872402 1.01499462 -0.2723291 1.00368721 0.724954717 1.07375920
## 379 -0.4872402 1.01499462 -0.2723291 1.00368721 0.135729129 0.98139312
## 387 -0.4872402 1.01499462 -0.2723291 1.25395112 -2.323647242 1.11638970
## 392 -0.4872402 1.01499462 -0.2723291 1.25395112 -0.332520192 0.49469492
## 394 -0.4872402 1.01499462 -0.2723291 1.19354259 -0.130418661 0.85350162
## 395 -0.4872402 1.01499462 -0.2723291 1.19354259 -0.565933227 0.92810499
## 400 -0.4872402 1.01499462 -0.2723291 1.19354259 -0.615746985 0.32772547
## 405 -0.4872402 1.01499462 -0.2723291 1.19354259 -1.072610303 0.59771863
## 418 -0.4872402 1.01499462 -0.2723291 1.07272553 -1.395688102 0.72916267
## 427 -0.4872402 1.01499462 -0.2723291 0.25289548 -0.637095738 -0.31528455
## 437 -0.4872402 1.01499462 -0.2723291 1.59914271 0.251012396 0.87836941
## 443 -0.4872402 1.01499462 -0.2723291 1.59914271 -0.093414156 1.11638970
## 450 -0.4872402 1.01499462 -0.2723291 1.36613839 0.188389387 1.05599649
## 453 -0.4872402 1.01499462 -0.2723291 1.36613839 0.017599361 0.82508129
## 454 -0.4872402 1.01499462 -0.2723291 1.36613839 1.577481596 1.09152190
## 456 -0.4872402 1.01499462 -0.2723291 1.36613839 0.342100410 0.63679658
## 457 -0.4872402 1.01499462 -0.2723291 1.36613839 -0.439263958 0.68653217
## 476 -0.4872402 1.01499462 -0.2723291 0.25289548 -0.174539418 1.02402361
## 478 -0.4872402 1.01499462 -0.2723291 0.51178918 -1.395688102 1.02047107
## 492 -0.4872402 2.42017014 -0.2723291 0.46864023 -0.429301206 1.07375920
## 498 -0.4872402 -0.21088983 -0.2723291 0.26152527 -0.698295497 0.07194248
## 500 -0.4872402 -0.21088983 -0.2723291 0.26152527 -1.018526795 0.17496618
## 502 -0.4872402 0.11562398 -0.2723291 0.15796779 0.438881424 0.01865435
## dis rad tax ptratio black
## 6 1.076671135 -0.7521778 -1.10502160 0.11292035 0.41016511
## 16 0.334118787 -0.6373311 -0.60068166 1.17530274 0.42659539
## 28 0.312653344 -0.6373311 -0.60068166 1.17530274 -0.55089661
## 34 -0.003724486 -0.6373311 -0.60068166 1.17530274 0.02295823
## 36 -0.206458943 -0.5224844 -0.76681717 0.34387304 0.44061589
## 43 0.914588047 -0.7521778 -1.03975408 -0.25660396 0.29241479
## 46 0.619913109 -0.7521778 -1.03975408 -0.25660396 0.44061589
## 49 0.985395514 -0.7521778 -1.03975408 -0.25660396 0.44061589
## 51 1.434032764 -0.6373311 -0.98041997 -0.76469989 0.42593818
## 52 1.434032764 -0.6373311 -0.98041997 -0.76469989 0.40852209
## 63 1.629121344 -0.1779443 -0.73715011 0.57482574 0.44061589
## 65 2.577684955 -0.7521778 -1.14062207 0.06672981 0.40052602
## 66 1.337533251 -0.6373311 -0.42267932 -1.08803366 0.44061589
## 77 0.122123795 -0.5224844 -0.06074124 0.11292035 0.18605612
## 78 0.140312434 -0.5224844 -0.06074124 0.11292035 0.33173792
## 88 -0.022530493 -0.7521778 -0.95668632 0.02053927 0.42144724
## 90 -0.180719408 -0.8670245 -0.82021787 -0.30279450 0.43141494
## 91 -0.333731922 -0.8670245 -0.82021787 -0.30279450 0.38891529
## 93 -0.061329756 -0.6373311 -0.82021787 -0.11803234 0.41991374
## 97 -0.142395045 -0.8670245 -0.78461740 -0.21041342 0.38508156
## 101 -0.483087712 -0.5224844 -0.14380900 1.12911220 0.41717537
## 110 -0.593596750 -0.5224844 -0.14380900 1.12911220 0.37850944
## 112 -0.530720099 -0.4076377 0.14099473 -0.30279450 0.42626679
## 116 -0.632538482 -0.4076377 0.14099473 -0.30279450 -0.12885754
## 118 -0.497524647 -0.4076377 0.14099473 -0.30279450 0.41443699
## 138 -0.923794146 -0.6373311 0.17066179 1.26768382 0.40972697
## 145 -1.163095840 -0.5224844 -0.03107419 -1.73470120 0.44061589
## 147 -1.082030551 -0.5224844 -0.03107419 -1.73470120 -2.05273356
## 155 -0.971474023 -0.5224844 -0.03107419 -1.73470120 -0.39053710
## 158 -0.910734419 -0.5224844 -0.03107419 -1.73470120 0.07400163
## 159 -0.967722319 -0.5224844 -0.03107419 -1.73470120 -0.03049494
## 171 -0.650204732 -0.5224844 -0.03107419 -1.73470120 -0.70523169
## 184 -0.450224689 -0.7521778 -1.27709053 -0.30279450 0.44061589
## 188 -0.003059627 -0.5224844 -0.06074124 -1.50374851 0.40742673
## 189 0.366459420 -0.5224844 -0.06074124 -1.50374851 0.28660942
## 199 1.668775425 -0.8670245 -0.47014661 -2.70470251 0.38913436
## 200 1.832330701 -0.7521778 -0.03700760 -0.67231881 0.44061589
## 202 1.175355183 -0.8670245 -0.35741180 -1.73470120 0.40633138
## 204 0.628271335 -0.6373311 -1.09315478 -1.73470120 0.39548740
## 207 0.265875775 -0.6373311 -0.77868399 0.06672981 0.41838025
## 214 0.071404563 -0.6373311 -0.77868399 0.06672981 0.31914137
## 215 -0.098561851 -0.6373311 -0.77868399 0.06672981 -0.08482439
## 228 -0.275129364 -0.1779443 -0.60068166 -0.48755665 0.16874956
## 229 -0.199430436 -0.1779443 -0.60068166 -0.48755665 0.22822717
## 230 -0.199430436 -0.1779443 -0.60068166 -0.48755665 0.25922563
## 231 -0.058670320 -0.1779443 -0.60068166 -0.48755665 0.23742812
## 235 -0.067978344 -0.1779443 -0.60068166 -0.48755665 0.03862176
## 236 -0.067978344 -0.1779443 -0.60068166 -0.48755665 0.21990249
## 239 1.137315760 -0.4076377 -0.64221553 -0.85708096 0.24903885
## 242 1.206746019 -0.4076377 -0.64221553 -0.85708096 0.41564187
## 249 1.914535748 -0.2927910 -0.46421320 0.29768250 0.19755732
## 254 2.427521836 -0.2927910 -0.46421320 0.29768250 0.44061589
## 255 2.576450217 -0.9818712 -0.55321437 -0.94946204 0.39669229
## 265 -0.885659742 -0.5224844 -0.85581834 -2.51994036 0.34192469
## 267 -0.789350190 -0.5224844 -0.85581834 -2.51994036 0.30008225
## 276 0.224274608 -0.6373311 -0.91515245 -0.39517558 0.44061589
## 279 0.163962412 -0.6373311 -0.91515245 -0.39517558 0.44061589
## 288 1.672669599 -0.4076377 -0.68374941 -0.85708096 0.44061589
## 290 1.672669599 -0.4076377 -0.68374941 -0.85708096 0.16480630
## 292 0.627653966 -0.6373311 -0.96855315 0.34387304 0.44061589
## 295 0.810965047 -0.6373311 -0.70748306 -1.13422420 0.44061589
## 298 1.199100142 -0.6373311 -0.70748306 -1.13422420 0.44061589
## 309 -0.226784628 -0.6373311 -0.61848189 -0.02565127 0.44061589
## 311 -0.605801659 -0.6373311 -0.61848189 -0.02565127 -0.06817505
## 317 0.096669199 -0.6373311 -0.61848189 -0.02565127 0.37270408
## 327 0.769743799 -0.5224844 -0.71934988 0.52863520 0.44061589
## 328 0.769743799 -0.5224844 -0.71934988 0.52863520 0.44061589
## 329 0.674146595 -0.6373311 0.12912791 -0.71850935 0.28222802
## 330 0.674146595 -0.6373311 0.12912791 -0.71850935 0.20303408
## 332 1.351400307 -0.9818712 -0.61848189 -0.71850935 0.40906976
## 333 1.351400307 -0.9818712 -0.61848189 -0.71850935 0.06107647
## 334 1.264826188 -0.5224844 -1.09315478 0.80577843 0.36186010
## 344 0.919906918 -0.5224844 -0.22687676 -0.39517558 0.44061589
## 347 2.003389383 -0.7521778 -0.33367816 0.15911089 0.08692678
## 351 2.373098390 -0.9818712 -0.43454615 0.57482574 0.44061589
## 362 -0.712131584 1.6596029 1.52941294 0.80577843 -0.06598434
## 371 -1.231243871 1.6596029 1.52941294 0.80577843 0.38749133
## 374 -1.244636028 1.6596029 1.52941294 0.80577843 0.44061589
## 375 -1.262302277 1.6596029 1.52941294 0.80577843 0.44061589
## 378 -1.157349560 1.6596029 1.52941294 0.80577843 0.44061589
## 379 -1.144004893 1.6596029 1.52941294 0.80577843 0.44061589
## 387 -1.105490570 1.6596029 1.52941294 0.80577843 0.44061589
## 392 -0.772776208 1.6596029 1.52941294 0.80577843 0.23775673
## 394 -0.951623237 1.6596029 1.52941294 0.80577843 0.44061589
## 395 -0.955944820 1.6596029 1.52941294 0.80577843 0.44061589
## 400 -1.089723917 1.6596029 1.52941294 0.80577843 -0.20279379
## 405 -1.038909706 1.6596029 1.52941294 0.80577843 -0.29808940
## 418 -1.019866249 1.6596029 1.52941294 0.80577843 -2.51179552
## 427 -0.853604048 1.6596029 1.52941294 0.80577843 -3.63683142
## 437 -0.851229552 1.6596029 1.52941294 0.80577843 -3.60572343
## 443 -0.850184774 1.6596029 1.52941294 0.80577843 0.42736214
## 450 -0.764607943 1.6596029 1.52941294 0.80577843 -0.57466575
## 453 -0.677606414 1.6596029 1.52941294 0.80577843 0.31125484
## 454 -0.637477434 1.6596029 1.52941294 0.80577843 0.21026340
## 456 -0.645503230 1.6596029 1.52941294 0.80577843 -3.34908249
## 457 -0.576737829 1.6596029 1.52941294 0.80577843 -3.79204278
## 476 -0.754635060 1.6596029 1.52941294 0.80577843 -0.59054835
## 478 -0.804641943 1.6596029 1.52941294 0.80577843 -0.07879996
## 492 -0.915103491 -0.6373311 1.79641644 0.75958789 0.36624150
## 498 -0.428521797 -0.4076377 -0.10227512 0.34387304 0.44061589
## 500 -0.662552110 -0.4076377 -0.10227512 0.34387304 0.42823842
## 502 -0.625177545 -0.9818712 -0.80241764 1.17530274 0.38683412
## lstat medv crime
## 6 -1.042290874 0.67055821 low
## 16 -0.585776111 -0.28626471 med_high
## 28 0.647934029 -0.84078708 med_high
## 34 0.797771697 -1.02562788 med_high
## 36 -0.416333515 -0.39499459 low
## 43 -0.958269752 0.30087663 med_low
## 46 -0.342114857 -0.35150264 med_low
## 49 2.542610329 -0.88427903 med_low
## 51 0.111599201 -0.30801068 med_low
## 52 -0.451342316 -0.22102678 low
## 63 -0.829437365 -0.03618599 med_low
## 65 -0.644590896 1.13809669 low
## 66 -1.117909883 0.10516285 low
## 77 -0.095652900 -0.27539172 med_low
## 78 -0.333712745 -0.18840782 med_low
## 88 -0.589977167 -0.03618599 low
## 90 -0.973673624 0.67055821 low
## 91 -0.538164142 0.00730596 low
## 93 -0.629187024 0.03992492 low
## 97 -0.183875078 -0.12316989 med_low
## 101 -0.452742668 0.54008236 med_low
## 110 0.405673128 -0.34062965 med_high
## 112 -0.349116618 0.02905194 med_low
## 116 0.435080520 -0.46023251 med_low
## 118 -0.329511689 -0.36237562 med_low
## 138 0.271239333 -0.59070837 med_high
## 145 2.329756820 -1.16697672 med_high
## 147 0.559711851 -0.75380318 med_high
## 155 0.345457990 -0.60158135 med_high
## 158 -1.129112700 2.04055467 med_high
## 159 -0.871447926 0.19214675 med_high
## 171 0.248833700 -0.55808940 med_high
## 184 -0.976474328 1.08373175 med_low
## 188 -0.836439125 1.02936681 low
## 189 -1.133313756 0.79016108 med_low
## 199 -0.844841237 1.31206449 low
## 200 -1.133313756 1.34468345 low
## 202 -0.731412722 0.17040078 low
## 204 -1.238340158 2.82340979 low
## 207 -0.235688103 0.20301974 med_low
## 214 -0.458344076 0.60532029 med_low
## 215 2.366165973 0.12690883 med_high
## 228 -0.881250390 0.98587486 med_high
## 229 -1.222936286 2.62769601 med_high
## 230 -1.245341918 0.97500187 med_high
## 231 -0.140464165 0.19214675 med_high
## 235 -0.644590896 0.70317718 med_high
## 236 -0.248291271 0.15952779 med_high
## 239 -0.881250390 0.12690883 med_low
## 242 -0.035437762 -0.26451873 med_low
## 249 -0.438739148 0.21389273 med_low
## 254 -1.276149663 2.20364948 med_high
## 255 -0.851842997 -0.06880495 low
## 265 -0.637589136 1.51865126 med_high
## 267 0.299246373 0.88801797 med_high
## 276 -1.354569377 1.02936681 med_low
## 279 -0.765021171 0.71405016 low
## 288 -0.772022931 0.07254389 low
## 290 -0.440139500 0.24651169 low
## 292 -1.273348959 1.60563516 low
## 295 -0.315508169 -0.09055093 low
## 298 0.446283337 -0.24277276 med_low
## 309 -1.136114460 0.02905194 med_high
## 311 -0.001829314 -0.69943824 med_high
## 317 0.794970993 -0.51459745 med_high
## 327 -0.910657783 0.05079791 med_high
## 328 0.019175967 -0.03618599 med_low
## 329 -0.375723306 -0.35150264 low
## 330 -0.744015891 0.00730596 low
## 332 -0.031236706 -0.59070837 low
## 333 -0.675398641 -0.34062965 low
## 334 -0.976474328 -0.03618599 low
## 344 -0.766421523 0.14865480 low
## 347 0.002371742 -0.57983538 low
## 351 -0.934463767 0.03992492 low
## 362 0.215225251 -0.28626471 high
## 371 -1.357370081 2.98650460 high
## 374 3.097149734 -0.94951696 high
## 375 3.545262384 -0.94951696 high
## 378 1.202473434 -1.00388190 high
## 379 1.545559682 -1.02562788 high
## 387 2.188321265 -1.30832556 high
## 392 0.855186130 0.07254389 high
## 394 0.352459751 -0.94951696 high
## 395 0.517701290 -1.06911983 high
## 400 2.424980758 -1.76499104 high
## 405 2.062289582 -1.52578531 high
## 418 1.958663532 -1.31919854 high
## 427 0.425278056 -1.34094452 high
## 437 0.755761136 -1.40618245 high
## 443 0.551309739 -0.44935953 high
## 450 0.932205492 -1.03650086 high
## 453 0.646533677 -0.69943824 high
## 454 0.572315020 -0.51459745 high
## 456 0.766963952 -0.91689800 high
## 457 0.890194931 -1.06911983 high
## 476 1.602974115 -1.00388190 high
## 478 1.716402630 -1.14523074 high
## 492 0.758561840 -0.97126294 med_low
## 498 0.202622083 -0.46023251 med_high
## 500 0.342657286 -0.54721641 med_low
## 502 -0.417733867 -0.01444002 low
## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2425743 0.2524752 0.2475248 0.2574257
##
## Group means:
## zn indus chas nox rm
## low 0.94980017 -0.9105920 -0.07145661 -0.8586919 0.5003426
## med_low -0.09440995 -0.2727685 0.03646311 -0.5432534 -0.1364047
## med_high -0.38347752 0.2464336 0.23949396 0.4255776 0.1250548
## high -0.48724019 1.0170690 -0.04518867 1.0400319 -0.4061323
## age dis rad tax ptratio
## low -0.8673785 0.8473704 -0.6818635 -0.7705710 -0.42345550
## med_low -0.3439487 0.3491847 -0.5472552 -0.4838749 -0.04014242
## med_high 0.5038605 -0.4020077 -0.3984500 -0.2761834 -0.24413252
## high 0.8078993 -0.8389260 1.6386213 1.5144083 0.78135074
## black lstat medv
## low 0.38063197 -0.78908436 0.585460441
## med_low 0.31523354 -0.11739954 -0.001861461
## med_high 0.09192049 0.03266136 0.166703960
## high -0.82414899 0.82202202 -0.666610184
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.12398859 0.709619648 -1.01674042
## indus 0.02212879 -0.244881780 0.20550238
## chas -0.06708134 -0.068704666 0.09084434
## nox 0.37776875 -0.717917520 -1.22801750
## rm -0.08385034 -0.103393572 -0.19194512
## age 0.31422520 -0.457667401 -0.32036627
## dis -0.07689234 -0.418110954 0.20665628
## rad 2.97555794 1.066402249 -0.11116982
## tax 0.01712221 -0.214966607 0.60561793
## ptratio 0.10867461 0.005481813 -0.31536610
## black -0.12511645 -0.005148864 0.11994751
## lstat 0.15954535 -0.166417054 0.51332882
## medv 0.15496084 -0.333619601 -0.14143962
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9423 0.0420 0.0157
## [1] low med_high med_high med_high low med_low med_low
## [8] med_low med_low low med_low low low med_low
## [15] med_low low low low low med_low med_low
## [22] med_high med_low med_low med_low med_high med_high med_high
## [29] med_high med_high med_high med_high med_low low med_low
## [36] low low low low med_low med_low med_high
## [43] med_high med_high med_high med_high med_high med_high med_low
## [50] med_low med_low med_high low med_high med_high med_low
## [57] low low low low low med_low med_high
## [64] med_high med_high med_high med_low low low low
## [71] low low low low low high high
## [78] high high high high high high high
## [85] high high high high high high high
## [92] high high high high high high high
## [99] med_low med_high med_low low
## Levels: low med_low med_high high
## predicted
## correct low med_low med_high high
## low 17 12 0 0
## med_low 3 16 5 0
## med_high 0 13 13 0
## high 0 0 0 23
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.2662 8.4832 12.6090 13.5488 17.7568 48.8618
While checking for the optimal number of clusters you should look in the graph when defining the k max = 10 for the k value where the curve varies the most dramatically. That appears to be around k = 2
When exploring the pairs using the colors to separate the km clusters, you can see that the variable that better separetes the two clusters (Red Vs Black cluster) would be tax (full-value property-tax rate per $10,000) and rad (index of accessibility to radial highways). An almost complete cluster separation can also be observed when you check the crim (per capita crime rate by town) and black (1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town) variables.
For the reasonable amount of clusters (> 2) I select k = 3 as it was the following position in the curve you could observe a dramatically change.
Now when you check at the pairs with the clusters divided by colors, since we have more clusters it gets a bit more confusing to read the data. Nonetheless, after careful observation you the variables were you can more easily identify the different clusters (Red, Black and Green) are medv (median value of owner-occupied homes in $1000s) and partialy successful will be age (proportion of owner-occupied units built prior to 1940).
I was trying the following approach but I guess I missed something, further work/learning required in this section ; if the peer review knows how to fix this and explain me what is wrong I highly appreciate it!!
k-means clustering km <-kmeans(boston_scaled, centers = 3)
linear discriminant analysis lda.fit <- lda(km ~ . , data = boston_scaled$km)
print the lda.fit object lda.fit
function for the lda biplot arrows lda.arrows <- function(x, myscale = 2, arrow_heads = 0.1, color = “red”, tex = 0.75, choices = c(1,2)){ heads <- coef(x) arrows(x0 = 0, y0 = 0, x1 = myscale * heads[,choices[1]], y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads) text(myscale * heads[,choices], labels = row.names(heads), cex = tex, col=color, pos=3) }
target classes as numeric classes <- as.numeric(boston_scaled$km)
plot the lda results plot(lda.fit, dimen = 2, col = classes, pch = classes) lda.arrows(lda.fit, myscale = 5)
I got stuck in this section as unfortunetely I couldn’t manage to fix the errors in my code
## [1] 404 13
## [1] 13 3
##
## Attaching package: 'plotly'
## The following object is masked from 'package:ggplot2':
##
## last_plot
## The following object is masked from 'package:MASS':
##
## select
## The following object is masked from 'package:stats':
##
## filter
## The following object is masked from 'package:graphics':
##
## layout
## Warning: 'scatter3d' objects don't have these attributes: 'col'
## Valid attributes include:
## 'type', 'visible', 'showlegend', 'legendgroup', 'opacity', 'name', 'uid', 'ids', 'customdata', 'hoverinfo', 'hoverlabel', 'stream', 'x', 'y', 'z', 'text', 'hovertext', 'mode', 'surfaceaxis', 'surfacecolor', 'projection', 'connectgaps', 'line', 'marker', 'textposition', 'textfont', 'error_x', 'error_y', 'error_z', 'scene', 'xcalendar', 'ycalendar', 'zcalendar', 'idssrc', 'customdatasrc', 'hoverinfosrc', 'xsrc', 'ysrc', 'zsrc', 'textsrc', 'hovertextsrc', 'textpositionsrc', 'key', 'set', 'frame', 'transforms', '_isNestedKey', '_isSimpleKey', '_isGraticule'